1. 1.1 – Limits: A Numerical and Graphical Approach
Function Review
𝑓 𝑥 =2𝑥−7
𝑓 𝑥 =2𝑥−7
𝑓 3 = ?
𝑓 −5 = ?
𝑓 3 =6−7
𝑓 −5 =2 −5 −7
𝑓 3 =2 3 −7
𝑓 −5 =−10−7
𝑓 3 =−1
𝑓 −5 =−17
3,−1
−5, −17

2. 1.1 – Limits: A Numerical and Graphical Approach
Function Review
𝑓 𝑥 = 𝑥 2 +2𝑥−7
𝑓 −7 = ?
𝑓 −7 = − −7 −7
𝑓 −7 =49−14−7
𝑓 −7 =28
−7, 28

3. 1.1 – Limits: A Numerical and Graphical Approach
Defn: Limit
As the variable x approaches a certain value, the variable y approaches a certain value.
lim 𝑥→𝑐 𝑓 𝑥 =𝐿
Find the requested limits from the graph of the given function.
lim 𝑥→2 𝑓 𝑥 = 3

4. 1.1 – Limits: A Numerical and Graphical Approach
Defn: Limit
As the variable x approaches a certain value, the variable y approaches a certain value.
lim 𝑥→𝑐 𝑓 𝑥 =𝐿
Find the requested limit of the given function.
𝑓 𝑥 = 𝑥 2 −1 𝒙
𝒇 𝒙
1.9
2.61
lim 𝑥→2 𝑓 𝑥
1.99
2.9601
1.999
2.9960
lim 𝑥→2 𝑓 𝑥 = 3 2 ?
2.001
3.0040
2.01
3.0401
2.1
3.41

5. 1.1 – Limits: A Numerical and Graphical Approach
Find the requested limits for the given function.
𝑓 𝑥 = 𝑥 2 −1
lim 𝑥→−1 𝑓 𝑥 =
−1 2 −1=
lim 𝑥→0 𝑓 𝑥 =
0 2 −1= −1
lim 𝑥→2 𝑓 𝑥 =
2 2 −1= 3

6. 1.1 – Limits: A Numerical and Graphical Approach
Given the following graph of a function, find the requested limit.
lim 𝑥→2 𝑔 𝑥 = 4
𝑔 2 = 6
2,6

7. 1.1 – Limits: A Numerical and Graphical Approach
Given the following graph of a function, find the requested limits.
lim 𝑥→0 𝑓 𝑥 = 2
𝑓 0 = 2
0,2
lim 𝑥→−1 𝑓 𝑥 =
𝐷𝑁𝐸
𝑓 −1 = 2
−1,2
lim 𝑥→1 𝑓 𝑥 = 2
𝑓 1 = 3
1,3
lim 𝑥→2 𝑓 𝑥 =
𝐷𝑁𝐸
𝑓 2 = 3
2,3

8. 1.1 – Limits: A Numerical and Graphical Approach
A limit of a function can be analyzed from the left and right sides of a particular value of x.
lim 𝑥→ 2 − 𝑔 𝑥 = 4
lim 𝑥→ 𝑔 𝑥 = 4
therefore
lim 𝑥→2 𝑔 𝑥 = 4

9. 1.1 – Limits: A Numerical and Graphical Approach
A limit of a function can be analyzed from the left and right sides of a particular value of x.
lim 𝑥→ 1 − 𝑓 𝑥 = 2
lim 𝑥→ 𝑓 𝑥 = 2 ∴
lim 𝑥→1 𝑓 𝑥 = 2
lim 𝑥→ −1 − 𝑓 𝑥 = 1
lim 𝑥→ −1 + 𝑓 𝑥 = 2 ∴
lim 𝑥→−1 𝑓 𝑥 =
𝐷𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡
(𝐷𝑁𝐸)

10. 1.1 – Limits: A Numerical and Graphical Approach
Given the graph of a function, find the requested limits.
lim 𝑥→ −3 − 𝑓 𝑥 = 3
lim 𝑥→ −3 + 𝑓 𝑥 = −1
lim 𝑥→−3 𝑓 𝑥 =
𝐷𝑁𝐸
𝑓 −3 = 1
lim 𝑥→ 0 − 𝑓 𝑥 = 2
lim 𝑥→ 𝑓 𝑥 = 2
lim 𝑥→0 𝑓 𝑥 =
𝑓 0 = 2 2
lim 𝑥→ 2 − 𝑓 𝑥 = 6
lim 𝑥→ 𝑓 𝑥 = −1
lim 𝑥→2 𝑓 𝑥 =
𝐷𝑁𝐸
𝑓 2 = 6
lim 𝑥→ 3 − 𝑓 𝑥 = 2
lim 𝑥→ 𝑓 𝑥 = −4
lim 𝑥→3 𝑓 𝑥 =
𝑓 3 =
𝐷𝑁𝐸 2

11. 1.1 – Limits: A Numerical and Graphical Approach
Given the graph of a function, find the requested limits.
lim 𝑥→ −3 − 𝑓 𝑥 = −1
lim 𝑥→ −3 + 𝑓 𝑥 = −2
lim 𝑥→−3 𝑓 𝑥 =
𝐷𝑁𝐸
𝑓 −3 = −1
lim 𝑥→ 0 − 𝑓 𝑥 = 2
lim 𝑥→ 𝑓 𝑥 = 2
lim 𝑥→0 𝑓 𝑥 =
𝑓 0 = 2 2
lim 𝑥→ 1 − 𝑓 𝑥 = 1
lim 𝑥→ 𝑓 𝑥 = −3
lim 𝑥→1 𝑓 𝑥 =
𝐷𝑁𝐸
𝑓 1 = −3
lim 𝑥→ 2 − 𝑓 𝑥 = −1
lim 𝑥→ 𝑓 𝑥 = −3
lim 𝑥→2 𝑓 𝑥 =
𝑓 2 =
𝐷𝑁𝐸 −3

12. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Infinity
Graphically
lim 𝑥→∞ 𝑓 𝑥 =
lim 𝑥→∞ 𝑔 𝑥 = ∞ ∞
lim 𝑥→−∞ 𝑓 𝑥 =
lim 𝑥→−∞ 𝑔 𝑥 = ∞ −∞

13. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Infinity
Graphically
lim 𝑥→ 1 − 𝑓 𝑥 =
lim 𝑥→∞ 𝑔 𝑥 = −∞
lim 𝑥→∞ 𝑓 𝑥 =
lim 𝑥→ 0 − 𝑓 𝑥 = 1 −∞
lim 𝑥→−∞ 𝑔 𝑥 =
lim 𝑥→ 𝑓 𝑥 = ∞ −1
lim 𝑥→−∞ 𝑓 𝑥 =
lim 𝑥→ 𝑓 𝑥 = ∞

14. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Infinity
Graphically
𝑓 𝑥 = 1 𝑥−1
lim 𝑥→∞ 1 𝑥−1 =
lim 𝑥→−∞ 1 𝑥−1 =
lim 𝑥→ 𝑥−1 = ∞
lim 𝑥→ 1 − 1 𝑥−1 = −∞
lim 𝑥→1 1 𝑥−1 =
𝐷𝑁𝐸

15. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Infinity
Numerically
𝑓 𝑥 = 1 𝑥−1
𝑓 10 = 1 10−1 =0.111
𝑓 100 = 1 100−1 =0.0101
𝑓 1000 = −1 =
lim 𝑥→∞ 1 𝑥−1 =0

16. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Infinity
Numerically
𝑓 𝑥 = 1 𝑥−1
𝑓 −10 = 1 −10−1 =−0.0909
𝑓 −100 = 1 −100−1 =−0.0099
𝑓 −1000 = 1 −1000−1 =−
lim 𝑥→−∞ 1 𝑥−1 =0

17. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Infinity
Numerically
𝑓 𝑥 = 1 𝑥−1
𝑓 1.01 = −1 =100
𝑓 = −1 =1000
𝑓 = 1 −1000−1 =10000
lim 𝑥→ 𝑥−1 =∞

18. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Infinity
Numerically
𝑓 𝑥 = 1 𝑥−1
𝑓 0.9 = 1 0.9−1 =−10
𝑓 .99 = −1 =−100
𝑓 = −1 =−1000
lim 𝑥→ 1 − 1 𝑥−1 =−∞

19. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Infinity
Graphically
𝑓 𝑥 = −1 𝑥+4 +3
lim 𝑥→∞ −1 𝑥+4 +3= 3
lim 𝑥→−∞ −1 𝑥+4 +3= 3
lim 𝑥→ −4 − −1 𝑥+4 +3= ∞
lim 𝑥→− −1 𝑥+4 +3= −∞
lim 𝑥→4 −1 𝑥+4 +3=
𝐷𝑁𝐸

20. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Infinity
Numerically
𝑓 10 = − =
𝑓 𝑥 = −1 𝑥+4 +3
𝑓 100 = − =
𝑓 1000 = − =
lim 𝑥→∞ −1 𝑥+4 +3=3

21. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Infinity
Numerically
𝑓 −10 = −1 − =
𝑓 𝑥 = −1 𝑥+4 +3
𝑓 −100 = −1 − =
𝑓 −1000 = −1 − =
lim 𝑥→−∞ −1 𝑥+4 +3=3

22. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Infinity
Numerically
𝑓 −4.01 = −1 − =103
𝑓 𝑥 = −1 𝑥+4 +3
𝑓 −4.001 = −1 − =1003
𝑓 − = −1 − =10003
lim 𝑥→− 4 − −1 𝑥+4 +3=∞

23. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Infinity
Numerically
𝑓 −3.9 = −1 − =−7
𝑓 𝑥 = −1 𝑥+4 +3
𝑓 −3.99 = −1 − =−97
𝑓 −3.999 = −1 − =−997
lim 𝑥→− −1 𝑥+4 +3=−∞

24. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Piecewise Functions

25. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Piecewise Functions
lim 𝑥→3 𝑓 𝑥 =
lim 𝑥→−2 𝑓 𝑥 =
lim 𝑥→3 2𝑥−1=
lim 𝑥→−2 − 𝑥 2 +4=
lim 𝑥→ −1=
lim 𝑥→−2 − − =
lim 𝑥→3 𝑓 𝑥 =5
lim 𝑥→−2 𝑓 𝑥 =0
𝑓 𝑥 = − 𝑥 2 +4 𝑥<1 2𝑥−1 𝑥≥1
lim 𝑥→1 𝑓 𝑥 =
lim 𝑥→ 1 − 𝑓 𝑥 =
lim 𝑥→ 𝑓 𝑥 = 1
− 𝑥 2 +4
2𝑥−1
lim 𝑥→ 1 − − 𝑥 2 +4=
lim 𝑥→ 𝑥−1=
lim 𝑥→ 1 − 𝑓 𝑥 =3
lim 𝑥→ 𝑓 𝑥 =1
lim 𝑥→1 𝑓 𝑥 =
𝐷𝑁𝐸

26. 1.1 – Limits: A Numerical and Graphical Approach
Limits Involving Piecewise Functions
lim 𝑥→−2 𝑓 𝑥 =
lim 𝑥→ −2 − 𝑓 𝑥 =
lim 𝑥→ − 𝑓 𝑥 =
lim 𝑥→− 2 − 𝑥+3=
lim 𝑥→ − 𝑥 2 =
lim 𝑥→− 2 − −2+3=
lim 𝑥→ − −2 2 =
lim 𝑥→ − 𝑓 𝑥 =4
lim 𝑥→− 2 − 𝑓 𝑥 =1
lim 𝑥→−2 𝑓 𝑥 =
𝐷𝑁𝐸
𝑓 𝑥 = 𝑥+3 𝑥<−2 𝑥 2 −2≤𝑥≤1 −𝑥+2 𝑥>1
lim 𝑥→1 𝑓 𝑥 =
lim 𝑥→ 1 − 𝑓 𝑥 =
lim 𝑥→ 𝑓 𝑥 =
lim 𝑥→ 1 − 𝑥 2 =
lim 𝑥→− −𝑥+2= −2 1
𝑥+3
𝑥 2
−𝑥+2
lim 𝑥→ 1 − =
lim 𝑥→ − 1 +2=
lim 𝑥→ 1 − 𝑓 𝑥 =1
lim 𝑥→ 𝑓 𝑥 =1
lim 𝑥→1 𝑓 𝑥 = 1

28. 1.1 – Limits: A Numerical and Graphical Approach